The present invention relates generally to techniques for measuring the interest rate exposure for fixed-income instruments, such as bonds.
A yield curve is a curve that shows the relationship between yields (i.e., interest rates) and maturity dates for a set of similar fixed-income instruments, such as bonds, at a given point in time. Yield curve exposure management is an important consideration that confronts many fixed-income portfolio managers. Portfolio managers typically use known analytical tools to precisely measure yield curve movements in order to effectively hedge their portfolio against systematic yield curve exposures (as opposed to idiosyncratic or non-systematic yield curve exposures).
Known statistical methods used by fixed-income portfolio managers to identify systematic changes in yield shape curve include principal component analysis (PCA) and factor analysis. Using these techniques, if N systematic changes in the yield curve are identified by either the PCA or factor analysis (referred to as “principal components” or “factors”), N hedge instruments can be used to hedge any bond (or other type of fixed-income financial instrument) against a combination of systematic curve moves.
Consider the example shown in FIG. 1. In this example, three principal components or factors (PC1, PC2 and PC3) have been identified. Each of the principal components (or factors) represent systematic reshapings of the yield curve. Two curves—symmetric about the 0 basis point change axis—for each principal component are shown in FIG. 1.
When three such principal components (or factors) are identified, three hedge instruments (such as futures), denoted as f1, f2 and f3, can be used to hedge one unit par-amount of a bond Y. In the following description, the dollar price change of fi relative to its base price, given an occurrence of PCj, is denoted as fij. This dollar price change may be referred to as the j-th factor exposure for hedging instrument (e.g., future) i. The dollar price change of Y, relative to its base price, given the occurrence of PCj is denoted as Yj. Thus, Yj may be considered the j-th factor exposure of bond Y.
The hedge weights (denoted as w1, w2, and w3) for each of the hedge instruments (f1, f2 and f3) to hedge the bond Y may be determined by solving the following set of equations:w1f11+w2f21+w3f31=Y1  Eq. 1 (hedge against occurrence of PC1)w1f12+w2f22+w3f32=Y2  Eq. 2 (hedge against occurrence of PC2)w1f13+w2f23+w3f33=Y3  Eq. 3 (hedge against occurrence of PC3)It should be noted that this approach assumes that any three hedge instruments can be used to hedge the systematic risk described by combinations of the three principal components.
A portfolio manager may compute the hedge weights for all bonds in the portfolio and the portfolio index. Typically, a maximum of three stable factors or principal components are estimated. A straightforward hedging approach would then be to use a set of three hedge instruments to hedge the yield curve exposure of all bonds in the portfolio and index. Picking the number of hedge instruments to match the number of principal components or factors is the simplest way of ensuring the existence of a unique solution.
While this hedging framework provides the benefit of parsimony, the drawback of this approach is that while the three principal components or factors are adequate for hedging systematic risk over the estimate interval, there is also some idiosyncratic risk in the yield curve. Further, the estimation of principal components or factors is dependent on the estimation interval; the factors change shape as the estimation interval is changed. This may be thought of as measurement error with regard to the factors.
Therefore, it is not generally the case that any combination of three hedge instruments will serve as equally good hedges for a given bond. Consider the case where the portfolio manager specifically wishes to underweight the exposure to the five-year sector of the yield curve, relative to the portfolio's index, by selling 5-year Treasury futures. In a simple three-hedging instrument framework, as discussed above, each bond in the portfolio and in the index is hedged by a combination of three hedge instruments. The portfolio and index interest rate exposure are expressed in terms of market weightings (e.g., w1, w2 and w3) in the three hedge instruments. The relative yield curve exposure is computed as the difference of the portfolio and index weightings.
Assume in this example that the 2-, 10- and 30-year futures are the hedge instruments. In this framework, therefore, the 5-year futures contract is indistinguishable from a combination of 2-, 10- and 30-year futures. For example, a $1 face amount in a 5-year future might be equivalent to a $0.68 face amount of the 2-year future plus a $0.66 face amount of the 10-year future minus a $0.12 face amount of the 30-year future in the hedging framework. In other words, given the estimation interval, the principal component approach assumes that over the long run, a position in 5-year futures is equivalent to a combination of 2-, 10- and 30-year futures. However, the portfolio manager has a specific trading view of the 5-year sector, which incorporates the idiosyncratic behavior of this sector. In addition, the portfolio manager's time horizon may be shorter than the long-run horizon of the PCA or factor model. Further, the principal components may change over time. Therefore, the manager would like its view to be expressed in the 5-year sector (as opposed to some combination of the 2-, 10- and 30-year sectors).
Accordingly, there exists a need for a greater degree of precision in the representation of yield curve exposure, while retaining some of the parsimony of the principal-component or factor hedging framework.